THE U N I V E R S I T Y OF M I C H I G A N COLLEGE OF LITERATURE, SCIENCE, AifD THE ARTS Departraent of Mathematics Technical Report No. 19 LOWER CLOSURE THEOREMS FOR LAGRANGE PROBLEMS OF OPTIMIZATION WITH DISTRIBUTED AND BOUNDARY CONTROLS David E. Cowles ORA Project 02416 submitted for UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR February 1971

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LOWER CLOSURE THEOREMS FOR LAGRANGE PROBLEMS OF OPTIMIZATION WITH DISTRIBUTED AND BOUNDARY CONTROLS* David E. Cowles I. INTRODUCTION In this paper we prove lower closure theorems for multidimensional problems of optimization with distributed and boundary controls. The concept of lower closure, introduced by Cesari [1,2,3,4] in connection with his existence theorems for optimal solutions, has the same role for Lagrange problems that Tonelli's lower semicontinuity has for free problems. The present analysis extends Cesari's theory in [3,4], but differs from it in two respects. First, we use the property Q(p), 0 < p < r+l, of upper semicontinuity of variable sets in E, which we introduced in a previous paper (D.E. Cowles [5]), instead of properties (U) (Kuratowski) and (Q) (Cesari) used in [3,4]. Property Q(p) reduces to property (U) for p = 0, and to property (Q) for p = r+l, as we proved in [5]. Also, for every 0 < p < r, property Q(p+l) implies property Q(p) (see our paper [5]). As in Cesari's analysis, we first prove a closure theorem (~ 2), which is then used to prove lower closure theorems (0 3,4). 2. A CLOSURE THEOREM Let G be a measurable bounded subset of the t-space E, v > 1, t = *Work done in the frame of US-AFOSR Research Project 69-1662. This is part of the author's Ph.D. thesis at The University of Michigan, 1970. 1

(t,...,t ). It is not restrictive to assume that G is a subset of the interior of the interval [-1,1], or -1 < t < 1, i = 1,...,v. We shall denote by -1 and 1 the points (-1,...,-1), (1,...,1) respectively. Analogously, for a = (a,..,aV) and b = (b,...,b with ai < b il,... we shall denote by [a,b] the interval [t E EV a < t < b, i = 1,...,v]. We shall denote by (t]R the set of all t = (t,..,t ) E with t,..., t rational. For every t E G let A(t) be a closed subset of the y-space E, y = (y,..,y ). Let A be the set of all points (t,y) E E x E with t E G, y c A(t). For every (t,y) e A let U(t,y) be a nonempty subset of the u-space Em, u = (u,...,um). Let M be the set of all (t,y,u) c E x 5 M E x E with (t,y) E A and u E U(t,y). For any subset F of G let AF, M denote the sets AF = (t,y) t e F, y c A(t)3 c A, M = ((t,y,u) t c F, y E A(t), u e U(t,y)} c M. Let f(t,y,u) = (fo, fl.. f2) be a continuous r+l vector function on r+l M, and for any point (t,y) E A let Q(t,y) c E denote the set Q(t,y) = Er+l z = f(t,y,u), u U(t,y We shall denote below by r(t), J(t), t E G, given measurable real valued functions on G, and by y(t), Yk(t), z(t), zk(t), uk(t), t c G, k = 1,2,..., given measurable vector functions on G as follows: 2

y(t) =(y, Y5), Yk(t)= (Ylk * Yk ) z(t) = (Z,...,z), z (t) = (z r.,z) Uk(t) = (" YU k )' for t E G and k = 1,2,.... We shall actually set all these functions V equal to zero in E - G, and we take Dk(t) = f Zk (t) dt for t E [-1, 1], k = 1,2,.... [-1,t] As in our previous paper [4] we denote by N (t,y ) the set of all (t,y) E A at a distance < 6 from (t,y ). For any (t,y ) A and 6 > 0 we denote by Q(t,y; 5) the set o 0 Q(t,Y;;) = U Q(t,y)O ~o (t,y) e N (t,y) Finally, if p if any integer, o < p < r+l, we say that the subsets r+l Q(t,y) of E have property Q(p) at a point (t,yo) E A provided for every o r r+l z = (z,...,z ) E, z O Ir~l i Q(t,y) Az E | z = z, i = p,...,r 00 J0 5~~~

= n cl co )Q(t y I E Er+l i 1-zi E> o >0 ooo' o o <, i = p,...,r3 We say that the sets Q(t,y) have property Q(p) on A if they have this property at every point (t,y ) e A. It is suitable to use the notations N(z; p) = E E z - z <, i = p,...,r Er+l i i N(z; p) = E Erl 1 = i As in [4] we shall say that the sets Q(t,y) have the upper set o 1 r property provided (to,Yo) e A, z = (z, z,.,zo) c Q(t,yo) implies that -o 1 r r+l -o s every other point z = (z z,...z ) E with z > z, is also a point of Q(t,yo) Lemma 2.1 Let G be measurable and assume that for every closed subset F of G the set MF is closed. Let f(t,y,u) = (f o,fl...,f )' (t,y,u) e M, be a 1 s continuous (r+l)-vector function on M, and let y(t) = (y,...,y ), t E G, be a measurable s-vector valued function an G with y(t) c A(t) a.e. on G. Then, for every measurable (r+l)-vector function E(t) = (E,...,Er), t E G, with E(t) e Q(t,y(t)) a.e. on G, there exists a measurable m-vector function u(t) = (u,...,u ), t E G, with u(t) E U (t,y(t)) and E(t) = f(t,y(t), u(t)) a.e. in G. 14

This lemma is a well known consequence of a McShane-Warfield theorem [7] and the proof, therefore, is omitted. Theorem 2.1 (a closure theorem). Let G be a bounded and measurable subset of E, for the sake of simplicity, say G c (-1,1). Let us assume that for every closed subset F of G the set P is closed. Let r(t) > O, t c G, be a given L-integrable function in G such that f(t,y,u) > - t(t) for all (t,y,u) E M. Let J(t), t E G, be a bounded positive measurable function -1 satisfying 0 < K < J(t) < K, t E G, for some fixed constant K. Let p be a given integer, 0 < p < r, let f(t,y,u) = (fofl,... f ) be continuous on M, and let us assume that the sets Q(t,y) have the upper set property and property Q(p+l) on A. Let us assume that y'yk. (L,(G))S, k = 1,2,... (2.1) i i Yk + Y strongly in L (G) as k ox, i = l,...,s, (2.2) Z, Zk (L G, k = 1,2,... (2.3) zk + Z weakly in L (G) as k +oo, j = 1,...,p, (2.4) Zk + zj strongly in L (G) as k +o, j = p+l,...,r, (2.5) k P )..

zk(t) = f (t,yk(t), uk (t)).J(t), j =,...,r, k = 1,2,... a.e. in G, (2.6) z~(t) = (f (t,yk(t), uk(t))+ W(t))J(t), k = 1,2,..., a.e. in G, (2. 7) Yk(t) E A(t), uk(t) E U(t,yk(t)), k = 1,2,..., a.e. in G.(2.8) Let D (t), t c [-1,1] be a monotone nondecreasing (in each variable) function of t with D(-1) = 0, and assume that Dk(t) + D (t) pointwise as k + oo for every t E [t n [ -1,1]. Let us assume that there is a decomposition D (t) = X(t) + S(t) of D (t) into two parts X(t) > 0, S(t) > 0 both defined on [-1,1] with X(t) = f z (t)dt, z0(t) > 0 on [-1,1], (2.9) [-l,t] z E L1 (G), z (t) = 0 on [-1,1] - G, and S(t) a singular function. Then y(t) e A(t) a.e. in G, and there is a measurable function u(t), t E G, such that u(t) E U(t,y(t)) a.e. in G, and z (t) = (f (t,y(t), u(t)) + *(t)) J(t) (2.10) z'(t) = fi(t,y(t), u(t)) J(t), i = l,...,r, a.e. in G.(2.11)

Proof. We shall first introduce suitable notations. For (t,y)eA, we define the following sets: + r+l Q(t,y) {z c Erl z = p + ( 0(t),0,...,) for p E Q(t,y)) and Q tJ(t,y) (-IZ = 1J(t) for p E Q (t,y)J. We will work with subsets CX, X = 1,2,..., of G. For t E C and 0, (t,y ) E A, define + +' (t,o'Y-E) U Q(ty) (t,y) C ANE (t,y ) with t E C and jCJC 0o1 C U 4r,JY) (t,y) C AwN(t,y ) with t E C For any interval [a,b] cE and any function z(t), t e Ev, we shall consider the usual differences of order v relative to the 2v vertices of [a,b], Az = A[ab] = z(b) - z(a) if v = 1 A = A zb]z = z(b,b ) -z (b,a ) -z(,a) + z(a,a )

if v = 2, and so on. Using this notation, we deduce from the pointwise convergence Dk(t) + D (t) for t E It)R, that for all intervals I = [a,b] c [-1,1], having rational coordinates AIDk = f zk(t) dt A aIDo(t) as k + o. Aik k I z Let t (t ",...t ) denote any point of G, let S = S (t ) denote the distance of t from the boundary of [-1,1], and let q = qh denote any closed hypercube q = (t = (tl... ) I tj < tj i + h, j = 1,2,...,v where tJ is rational, j = 1,2,...,v, h is a positive rational with 0 < h < 6 /v and t c q. By differentiation of multiple integrals and the o 0 definition of a singular function we have lim hV fy (t)dt = y (t ), i = 1,2,.,s, (2.12) h+-O q lim hV fz (t)dt = z (t), i = 1,..,r, (2.13) h+O q lim h- fz (t)dt = z (t (2.14) h+0 q lim h-v A = 0 (2. 15) h0+O for almost all t E G.

For almost all t e G, we have (t,yk(t)) E A for all k = 1,2,.... The convergences Yk(t) + y(t) in (L (G))S and zk(t) - z (t) in L (G), i =P + 1,...,r, as k + oo, imply convergence in measure on G, and hence there is a subsequence [Yk (t)], [zk (t)] which converges pointwise almost P P everywhere in G as p + o, i = P + 1,...,r. For simplicity of notations we denote this subsequence still [k]. Let G be the set of all t E G where 0 relations (2.12) through (2.15) hold, where (t,yk(t)) E A for all k and where lim yk (t) = y(t), lim Zk (t) = z(t) i = p +,..,r.!k->o k k+o (2.16) We see that G is measurable and iG I = IGI. Since A(t) is a closed set, Yk(t) + y(t) and Yk(t) E A(t) for t E G, we have y(t) E A(t) for t E G, that is, y(t) c A(t) almost everywhere in G. Because of the pointwise convergences (2.16) on G with IG I = IGI, we know that there are closed sets C, k = 1,2,..., with C G, C CX+ C1, ICxI > IGI - X- such that y(t), Yk(t), z k(t) z (t ), and r(t) are continuous on Ck, i = p + 1,...,r, k = 1,2,..., limits (2.16) take place uniformly on Ck as k + o, and this holds for every X = 1,2,.... Since G is bounded, each set Ck is compact and hence yk(t),y(t), z (t) k and z (t), i = p + l,...,r, are equicontinous and uniformly continuous on each C. Let x be any fixed integer, X~ > |GI G; hence ICkI >o. Let ~ > O be an arbitrary positive number. There exists some b' = b'(E,\) > O such that O o

It-t'I < <' with t,t' E Cx implies ly(t) - y(t')l < and!yk(t) - Yk (t')I < E for every k = 1,2,.... Also there exists some k = k (e,x) such O O that k > k (E,X), t E C, implies ly(t) - Yk(t) I < c. Let AN, MX denote the sets A AC =(t,y)|t E Cx, y E A(t)) cA, and M = M =(t,y,u)| x CC (t,y,u) E M, t E C). Let X (t) and )Q(t) be the characteristic functions of the sets CX and [-1,11 - CX so that X (t) + X*(t) = 1 for t c G. All X (t) and X*(t)zi(t) are in L (G) for i = 1,2,...,p. For every t E Ch we have X(t ) = 1 and X*(t )z (t) = 0, i = 1,2,...,p. Then for almost all t E C we have lim + hV f X (t)dt = im Iqn A C /q = 1 (2.17) h-+O q h+0 lim + hV X(t) zi(t) dt = (2.18) h+O q where i = 1,2,...,p. Let C' be the subset of Cx where (2.17) and (2.18) occur. Let H and H* be the sets H = q n C~ and H* = q - H. Then C~ is measurable, CK c Cx c G C G and ICkI c = CI l > 0, CQI = |G I = IGI~ X=l Let 1 > 0 and B > 0 be any positive numbers independent of e. Let t 0 be any point of C', and set yo = y(t ) and M = max{Ix (t )f + 1), where the maximum is taken over i = 0,1,...,p. Let us fix h so small that 0 < h < E/v, h < S /y, h < 5' and also so o o small that 10

z(t ) - h-v f z (t) dt| 9 min (T(r+l),1), i = 0,1,.**p (2. 19) |1 -(Iql/IHI)l <min (n(r+l) M11), (2.20) |h QA S| I r(r+l)~' (2.22) sup z (t) - z(t ) min (r+l)-1,, i = p +,..,r, tEq n c. (2. 23) This is possible because of relations (2.12-18). For any integer k > 0, let zk(t) be the (r+l)-vector function zk(t) 0 1 r -( (zt)z (t)..z (t)), t E G. k k k For t e H and k > k (e,X) we have 0!t-t I s vh 5 min fy(Ekt' k(t) - y(t )i Yk(t) - Yk(to) + IYk(to) - Y(t) + = and hence (t,k (t)) E N (to o ) for t e H and k > k (e,)). For all t c H we have, therefore, z (t) 6 Q~ (t,Y,)3), k > k (E,%). (2.24) k,J,Co 0 0 0 The hypothesis of weak convergence of zik(t) to z (t), as k + o implies 11

that lim f z (t)dt = z (t)dt, i = 1,2,...,P. k-Jv H H We can determine an integer k' = k'(t,e,X,jB ), k' > k (e ), such that for 0 0 k > k' we have sup lzk(t) - z(t)I - min (q(r+l),J, i = p + 1,...,r, (2.25) tcH Ifzi (t) dt- fz (t) dtl 5 (r+l) -1IHI, i = 1,...,p. (2.26) Hk H Now for k 2 k'(t,e,k,q ), and i = 1,2,...,p, we have 0 i 1 Iz (t ) - HI f zi(t) dtl I 0 Hk q q I JHK f(z (t) - z1(t)) dtl + I I|H J z(t) dt I H k H* 1 2 +d3 +4. (2.27) By (2.19) we have dl 5 q(r+l), by (2.26) we have d3 5 r(r+l), and by (2.21) we have d4 s q(r+l) -1 Also d2 = I 1- Iql(IHI)-l I ql fzzi(t) dtl -1 -1 s (r+l) M1 M11 by the definition of M1, (2.19) and (2.20). Thus (2.27) yields for i = 1,2,...,p, 12

Iz (t ) - HI f zk(t) dtI S 4(r+)- 1. (2.28) H k For i = p + 1,...,r, we have |z (t ) - |H f z(t) dtl II HI| f(Z (t) - z k(t)) dt H H S Up fzi(t) - zi(t) (2. 29) Using (2.25) and (2.23) we have for i = p+l,. fz (t ) - I HI i (t) dtl 5 2(r+l)- 1 (2.30) For i = O, because of assumption (2.7), h vf z(t) dt > O for all k. (2.31) We also have -v 0 v h f Zk(t) dt h A D (2.32) q q(k and h V z (t) dt =h A D. q On the other hand, Do = X + S and by (2.22) h- D h-a X + h-VA S (233) AqO A Iq q

with Ih-vA S| (r+l) rp. Also, since A D approaches A qD0 as k approaches infinity, q c G having vertices with rational coordinates, and we can determine k'(t,c,\,rl,) above so that for k > k' we have also 0 Ih D- h A D I (r+) (2.34) qk q0 Finally, (2.32), (2.33) and (2.34) yield IhV fz (t) dt - h fz (t) dtl = hVA - h -vA XI qk q qk - h- D - h-v D~ I + I h-v S{ s 2(r+)- (2 35) We have 0 10 0 -I IHI1 J (z(t) - z (t)) dtl q 1 0 + |H| fz O(t) dt = d + d6 + d + d8 (2.36) H* k 5 6 7 8 By (2.19) we have d5 -r (r+l), by (2.20) and (2.35) we have d -4(r1, and by (2.31) we have d8 Also, 14

d = -I(l-Jql((IHI)-z )IIqf- fz~(t) dt > -(r+l)- lI, ~~~~6 q where we have used (2. 20), (2. 19), and the definition of M1. Hence, (2.36) yields 0 1 0 z (t ) - HI fzk(t) dt 2 -6r(r+1i)- (2. 37) We have also IZ(t ) - Hj1 fz (t) dtl 4r(r+l) (2.28) i = 1,2,...,p, and i 1 Iz (t) - IH1 fz'(t) dtl S 2r( r+l), (2.30) i = p + 1,...,r. Equations (2.23) and (2.25) imply that zk(t) e N2 (z(t); p + ) for t e H. (2.38) Because of (2.24) and (2.38) |H| fz (t) dt E clco (Q (t,y,3) n N (z(t ); P + 1)). Hk O The latter set has the upper set property by statements (3. i), (3. ii) and (3.iii) of [4]. From (2.37), (2.28), and (2.30), 15

z(t) e (clco(Q, oyo30 n N2((Z(t)); p + 1))12l for every r > 0, P > 0, and E > O. Since r is an arbitrary positive number and the set inside the parenthesis is closed, z(to) E ClCO (Q, J C (to,Yo,3E) n N2(z(t); p +1)) for every E > 0 and every P > C. By statements (2.v) and (2.vi) of [4], the set Q j(ty) has property Q(p+l) on Ak. Therefore, z(t) Q, (t and z(t)(J(t)) - ((t),O,...,0) e Q(t,yo) for t E C But I U C = jG|. Therefore, z(t)(J(t)) - (*(t),O,...,O) E Q(t,y(t)) for almost all t E G. The conclusion of the theorem now follows from lemma (2. 1). 3. A PRELIMINARY LOWER CLOSURE THEOREM We shall use the same notations as in 9 2, in particular let f(t,y,u) denote a vector function f(t,y,u) = (f,f f ) defined on M. Here we shall consider the functional I [y,u] = J f (t,y(t), u(t)) J(t) dt. G o 16

Instead of the sets Q(t,y) of ~ 2, we shall consider here the sets Q(t,y) = z = (z,z,...,z ), z > f (tyu), i r+l z fi(t,y,u), i = l,...,r, u c U(t,y)} cE defined for every (t,y) c A. Theorem 3.1 (a lower closure theorem). Let G, A(t), A, U(t,y), M, and M be defined as in ~ 2, G measurable, A closed, M closed for every closed subset F of G. Let f(t,y,u) = (f,fl,...,f ) be continuous on r M, let p be any integer, 0 < p < r, and let us assume that the sets Q(t,y) have property Q(p+l) on A. Let f(t), J(t), t ~ G, be measurable -1 functions real valued on G with 0 < K < J(t) < K for all t E G and 1 s 1 s some constant K. Let y(t) = (y,... ), Yk(t) = (Yk k), z(t) k~1,~~~,Zr), Zlk(tt = ( k k z,...,z ), z (t)= (Zk,..Z Z), (t) = (uk,... ), t E G, be as in Theorem (2.1), k = 1,2,..., satisfying (2.1-6) and (2.8-9), and let us assume that limk~ I [Yk,uk] = a < +. Then y(t) E A(t) ae. on G, and 1 m there exists a measurable function u(t) = (u,...,u ), t E G, such that u(t) E U(t,y(t)), z (t) = f(t,y(t), u(t)) J(t), i =,...,m, ae. on G, and I [y,u] < a. Proof First set D (t), t e [-1,1], equal to k(t) = (fO (tyk(t),uk(t)) + (t))J(t) dt, k[-l,t ] 1 17

and let e be the real number, e = f J(t)r(t) dt. We may assume without loss [-1,1] of generality that 0 - Dk(t) - a0 + e + 1. Then using a diagonal process, we may extract a subsequence of the original sequence, say still k = 1,2,..., so that Dk(t) converges pointwise to a number Do(t) for each point t e [-1,1] having all rational coordinates. We have defined D (t) on the rationals. For a point t = (t,...,tV) E [-1,1] having at least one irrational coordinate we define D (t) = sup D (t), where sup is taken over all t = (t,..,V), -i -i i with t rational, t < t, i =,...,v. Hence, Do(t) is defined on [-1,1]. Also, since the functions Dk(t), k = 1,2,..., are nonnegative, monotone nondecreasing (in each variable) and equibounded, Do(t) is nonnegative, monotone nondecreasing and bounded. Since D (t) is of bounded variation, we may, using the Lebesgue decomposition theorem, decompose DO(t) as Do(t) = X(t) + S(t), where 0 X(t) = f zO(T) dT, t E [-1,1] [-l,t] S(t) > 0 is singular and monotone decreasing, and z (t) > 0 is L-integrable in [-1,1] and zero on [-1,1] - G. We now set up an auxiliary problem to which we will apply theorem 2.1. In this situation, A(t) and A are defined as above. We define the set U(t,y) as ~ 0 U(t,y) = [(u,u) E ]lu~ 2 fo(t,y,u), u E U(ty)) for (t,y) c A, and we define the set M as M- (t,y,u) I (t,y) E A and u E U(t,y)) 0 0 = ((t,y,u,u) I (t,y) c A and u 2 fo(t,y,u) 18

for u E U(t,y)). For any subset F of G, the set M is defined as the set of (t,y,u) e M for which t E F. For each closed subset F of G, since fo is continuous and MF is assumed to be closed, MF is closed. 0 Let f(t,y,u) = (u,fl(t,y,u),...,f r(t,y,u)). r+l The set Q(t,y) = [z E E E z = f(t,y,u), u e U(t,y)J has the upper set property, and we have assumed that it also has property Q(p+l) on A. Let y(t), yk(t), z(t), zk(t), J(t) and r(t) be as in the statement and proof of this theorem, with -Z(t) = ('Z(t), Z (t),... k(t)), O o z (t) = (uk(t) + *(t) ).J(t) k = 1,2,..., Let uk(t) be the control vector for the auxiliary problem defined by k(t) = (fo(t'yk(t),uk(t )),uk(t)), k = 1,2,.... We have Dk(t), Do(t), X(t) and S(t) defined as above. Relations (2. 1-6), (2.8-9) of Theorem (2. 1) hold by hypothesis, and we have just verified (2.7), as well as the pointwise convergence Dk(t) + D (t) as k + o for every t = (t,...,t ), t [-1,1], with rational coordinates. Thus, theorem (2.1) holds in the present situation. Hence, y(t) e A(t) ae. on G, and there is a measurable function u(t) = (u,u) 19

01 m (u,ul...,u ), t c G, such that u> f (t,y)t),u(t)), u(t) C U(t,y(t)), z (t) = f(t,y(t), u(t)) J(t), i = 1,...,r, and z0(t) = [u (t) + p(t)] J(t) ae. on G. We set u (t) = 0 on [-1,1] - G, and then, because of S(1) > 0, we have f u (t)J(t)dt = f (z (t) - *(t)J(t))dt [-1,1] [-1,1] = X(1) - e = ( X(1) + S(1) ) - (e + S(1) ) = a0 + e - ( e + S(1) ) = a0 - S(1) - a0. Because f0(t,y(t),u(t)) is a continuous function of three measurable 0 functions, it is measurable. Also, J(t) and u (t) are integrable and we have -g(t) < f (t,y(t),u(t)) < u~(t), and K1 < J(t) < K ae on G. Thus, f (t,y(t), u(t)) J(t) is integrable on G, and I[y,u] = J f (t,y(t), u(t)) J(t) dt G -Jf f (t,y(t), u(t) J(t) dt < S u (t) J(t) dt [-1,1] [ -i,1] < a. - o 20

Theorem 3. 1 is thereby proved. 4. A LOWER CLOSURE THEOREM FOR OPTIMIZATION PROBLEMS WITH DISTRIBUTED AND AND BOUNDARY CONTROLS In this section we state and prove a lower closure theorem for optimization problems with controls, state variables, constraints, and state equations on both the domain and its boundary. We shall first introduce a few definitions. We begin with C. B. Morrey's definition of regular transformations of class K from his paper [8]. Let T and S be subsets of Euclidean spaces. A transformation x = x(y) of T onto S is said to be of class K provided it is one to one and continuous, and the functions x = x(y) and y = y(x) satisfy a uniform Lipschitz condition on each compact subset of T and S respectively. The transformation is said to be regular if, in addition, the functions x(y) and y(x) satisfy a uniform Lipschitz condition on the whole of T and S respectively. Let G be a bounded measurable subset of EV, v > 1, whose boundary will be denoted by aG. Let F., j = 1,2,...,N, be subsets of aG, each of which is the image under a regular transformation t. of class K of a bounded interval R' of J U E-. Let r be a closed subset of and let kt be a measure defined J=l j on Uj Prj. For each j = 1,..,N, we assume that if e is a subset of Fj, -1 measurable with respect to Il, then E = t (e) is measurable with respect 21

to Lebesgue (v-l)-dimensional measure I | on R'.. Also, we assume the J converse, so that measurable sets on r. and R' correspond under t., J j j = 1,2,...N. We assume that there is a constant K > 1 such that if e = t.(E) is t-measurable, then J K Ejl [i(e) - KjEj (4.1) independent of j = 1,...,N. Since 1, induces a measure on each set R'. J via the transformation tj, j = 1,2,...,N, we may define J.(t), t e R', as the functions in L (R') which satisfy the equations J p(t.(E)) f J.(t) dt (4. 2) J E for every measurable subset E of R', j = 1,2,...,N. For every t E cl(G), let A(t) be a nonempty closed subset of y-space E. Let A be the set of all points (t,y) with t E cl(G) and y c A(t). For every (t,y) c A, let U(t,y) be a nonempty subset of u-space Em. Let M be the set of all (t,y,u) e E x E x E such that (t,y) E A and u e U(t,y). o s For every t E r, let B(t) be a nonempty closed subset of y-space E. Let B be the set of all (t,y) with t E r and y E B(t). For every (t,y) e B, let V(t,y) be a nonempty closed subset of v-space E. Let M be the set of all (t,y,v) C EV x E x E with (t,y) E B and v c V(t,j). Let f(t,y,u) = (fo, fl',rf ) be a continuous (r+l)-vector function on M, and let us consider the sets 22

(ty) { (~0 r r+l 0 > f Q(t,y) z =(z 1,...,z) C E Iz f(ty,u), (z 1,...) = (f"1,''f )(ty, u), u c U(t,y)) (4.3) Let g(t,y,v) = (go0,. gr,) be a continuous (r'+l)-vector function on M, and let us consider the sets Rt, Y o r' r'+l O ~ R ) E E z g0(t9,,v) r' (z,... ) = (gl",gr')(t2y,v) v E V(t,y)) (4.4) We assume that there are two functions 4(t), t e G and ~(t), t e r, such that fo(t,y,u) 2 -r(t) for all (t,y,u) in M, *(t)' O, 4(t) E L1(G), and go(t,yv) — r(t) for all (ty,v) in, (t) O v(t) c L(r). We consider here the functional 0 o I[y,y,u,v] f f (t,y(t),u(t)) dt + f g (t,y(t), v(t)) dt. G F In the lower closure theorem below we shall deal with sequences of functions all defined on G and r: zyt )Z(t) = (z1,.. ), k(t) =Zk) 1 s 1 s Y3) y'99y(t) = ((y...9 y ),y k yk (Ykt =k...,y)k k) 1 m uk(t) = (u,...,uk), t E G, k = 1,2,... ol or ol or z(t) = (z,.z ) zk(t) =(Z k"' Zk" Zk )' 23

o0 ol oS 0ol oS1 y(t) = (y, y Yk(t) =(Yk Yk )' 1 m' vk(t) = (Vk,. vk t E r k = 1,2,.... Theorem 4.1 (a lower closure theorem). Let G be bounded and measurable, o 0 0 A, B, M, M closed, f(t,y,u) continuous on M, g(t,y,v) continuous on M, and assume that for some integers p, p', 0 < p < r, o < pt' rt, the sets Q(t,y) have property Q(p+l) on A, and the sets R(t,y) have property Q(p'+l) on B. Let us assume that there are functions *(t) > O, t E G, I E L (G) and 0 0 *r(t) > O, t E r, * c L1 (r), such that f (t,y,u) > - *(t) for all 0 0 0 (t,y,u) E M, and g (t,y,v) > - (t) for all (t,y,v) E M. Let us assume that the functions z (t), z ( ), y(t) (t) i =... k k j = 1,...,s, are in Ll(G), that the functions ua(t) are measurable on G, j = 1,...,m, that f (t,yk(t), uk(t)) ~ L (G), and that yk(t) E A(t), uk(t) E U(t,yk(t)), zk(t) = fi (tk(t), uk(t)) a, e. on G, k = 1,2,.... (4.5) Oi 0o Let us assume that the functions z (t), z(t), y(t), (t), i k Yk =,...r', j l,...,s, are in LL(r), that the functions Ak(t) are j k1 o measurable in r, j = 1,...,m', that g (t,yk(t), vk(t)) c L1 (r), and that oi Yk(t) E B(t), vk(t) E V (t,Yk(t)), zk(t) = gi(tYk(t),vk(t)) 24

i.-a.e. on r, k = 1,2,.... (4. 6) Finally, let us assume that as k oo we have zk(t) -+ z (t) weakly in L1(G), i = 1,...,p, (4. 7) i i zk(t) + z (t) strongly in L (G), i = p+l,...,r, (4.8) k 1 yk(t) 7 yJ(t) strongly in L (G), j = l,...,s, (4.9) 01 Oi zk t) + z (t) weakly in LL(F), i = 1,..,p', (4.10) ~i ~i zk(t) + z (t) strongly in L(), i = p'+l,...,r', (4.11) y3(t) + yJ(t) strongly in Ll(r), j = 1,...,s', k 1 l'mk+ I [yk,yk, uk Vk] < a <+ (4.12) Then, y(t) E A(t) a.e. on G, y(t) E B(t),i-a.e. on r, and there are measurable functions u(t) = (u,...,m ), t E G, and ut- measurable functions v(t) = (v,...,v ), t c r, such that f (t,y(t), u(t)) c L,(G), go(t,y(t), v(t)) e L1 (r), and such that u(t) E U(t,y(t)), z (t) = fi(t,y(t), u(t)), i = 1,...,r, a.e. on G, 0 O 0 on r, I [y,y,u,v] < a. 25

Proof: We may write I(ykykuk vk) = f fo(t,y(t),u (t)) t + G k k N E J g (tj(z), Yk(tj(t)), vk(tj(t))) Jj(t) dt, j=l A where -1 k n j = cl t [r n (r- il k = 12,... We may assume that I[yk,yk,ukvVk] ao + 1. Since f (t,y,u) 2 -.r(t) for all (t,y,u) in M, the integrals of f on G are uniformly 0 bounded below by the number f-4(t) dt. Also, because of the fact that go(t,y,v)' — (t) and J(t) 5 K, the integrals of g on Aj are uniformly bounded below by the numbers f,, 4-(t (t)) K dt, j = 1,2,...,N. Hence, each of the integrals in I(Yk,Yk,uk,vk) on G and Aj, j = 1,2,...,1 is uniformly bounded above and below. We may, assume, therefore without loss of generality, that a g (tj(t), Yk(tj(t)), vk(tj())) J(t) dt approaches a finite limit aj, j = 1,2,...,N, as k approaches infinity. 26

We then have N lim f fo(t,yk(t), uk(t)) dt a - E a.. 0 lykkt)' Uk~t0o j=l k~ G o We shall apply lower closure theorem 3. 1 on each set A c R' 3- 3 j = 1,2,...,N. Here we have B(t) - B(t.(t)), t E A., which is a O3 3 nonempty closed subset of y-space E. Let Bj, j = 1,2,...,N, be the set of all (t,y) with t e Aj and y E B(t). For every (t,y) E Bj, let V(t,y) 0 0 be defined as the set V(t,y) V(tj(t),y). Let Mj, j = 1,2,...,N, be the set j = ((t,y,v) (t,y) c Bj and v E V(t,y)J. Since M is assumed to be closed and A is closed, Mj is a closed subset V s m or E xE x E o The function g(tj(t), y, v) = (g'''g,)(tj(t),yv) is a continuous (r'+l)-vector function on M.. Let R.(t,y) be the set 3 3 {z Erl E z >2 g(t j(),y,v), (z,...,z) = (g,..,g).) (tj(t),y,v), for v E V(t,y)) for (t,y) c Bj. 3 3 Then, we have Rj(t,y = R(tj(t),y) for (t,y) Bj, j 1,2,...,N. Now if (t',y)e NE(t,y)nB, then (tj(t'),y) is in N+- (t,%) B. where K is the Lipschitz constant of the transformation tj and t = tj(t). Hence, for each j = 1,2,...,N, and (t,y) E Bj, 27

R.(t,y,) R(tj(t),y,K c + e), and 3 3 n n cico (Rj(t,y, ) n N(zo; p +1)) C>O C>0 is a subset of n n clco(R(t(tt), y, K E + E) n N (z; p' + 1)). We see that, since R(t,y) has property Q(p'+l) on B, the set R(t,y) has property Q(p'+l) on Bj, for each j = 1,2,...,N. We have (tyk(tj(t))) B and vk(t (t)) in V(t (t), y(t(t))), a.e. on j,,2,,N, and k 1,2,.... For j =,...,, i = l2,...,rN, and k = 1,2,..., let us take 01 ~ i oi O z (j; t) = z (tj(t)).J(t), z (j; t) = zk(tj(t)).J(t). By virtue of the convergence relations (4. 10), (4. 11), and the relation K |El 5 g(e) - KjEJ, e = t.(E), we have 0 < K-1 J (t) - K, a.e. on A. J 3 i(j;t) + Z(j;t) weakly in L(Aj) i = 1,2,...,p k zk(j;t) i z (j;t) strongly in L(Aj), i ='+l,..,r' j = 1,2,...,N, as k approaches ~. Finally, r(tj(t)) is in Ll(Aj), 28

(tj(t)) 02 O, and g(tj(t ),y,v) - r(tj(t)) for all (t,y,v) in Mj, j = 1,2,...,N. Applying lower closure theorem, 3.1 we see that y(t.(E)) E B(t (t)) and J. there are measurable controls v (t), such that v (t) r Ci such that v. () c v(Z (t zi(tj(t)) = gi(t. (t)), v(t)), and g (t, y(t), vj(t))di < a, j = 1,2,...,N. Setting v(t) = vj(t. l(t)) on t.(A.), we see that there is a measurable control v(t), t c r, such that y(t) E B(t), v(t) E V(t,y(t)), z i(t) = gi(t,y(t),v(t)) a.e. on r, and O N f g0 (t,y(t),v(t)) d-i E aj. F j=l On G itself, we have exactly the situation of the lower closure theorem with J(t) = 1. Therefore, y(t) E A(t), a.e. on G, and there exists a measurable control u(t), t E G, with u(t) E U(t,y(t)), z (t) = f.(t,y(t),u(t)), a.e. on G, i=l,...,r, N J fo(t,y(t),u(t) dt S a - Z a. G jl The conclusion of theorem 4. 1 follows from the conclusions of this and 29

the preceding paragraphs. Remark 1. Suppose that all of the hypotheses of theorem 4. 1 hold except that I(y,y,u,v) is written as I(y,y,u,v) = S f (t,y(t),u(t)) dt + G f g0(t,y(t),v(t))d. + T(y(t),y(t)) r We see that we could have proven the same lower closure theorem for I(y,y,u,v) provided that T(y,y) < lim T(yk,yk) and f f (t, (t,Y),uk(t)) dt keo k G + f g (t,Yk(t),vk(t)) d.i approaches a finite limit as k + co. Remark 2. We mention a variant of theorem 4.1. We may assume that G and r are made up of a finite number of components Gl,...,Gd and 1....rd, and that, in each of these, there is a different system of control equations similar to the ones on G and r in theorem 4. 1. Also, we mention that the sets r. throughout this paper are thought of as subsets of the boundary aG of G because this will be the main application we have in mind, but actually the sets Fj could be subsets of G instead, or even abstract sets in no way connected with G. Examples The following two examples illustrate the use of the intermediate properties Q(p), O < p < r, used in connection with lower closure theorems in the present paper. Both examples have been mentioned already in [5].

Example 1. Let us consider the problem of the minimum of the cost functional I[x,ul,u2,v]= ff(2 +X +2+2 +u 22+u 2)ddy G with differential equations x< - U, xX U2, a.e. in G, and boundary conditions yx = v s a.e. on r = aG, 2.2 where G = [(%,y)2 + l < 1], r is the boundary of G, s is the arc length on F, yx the boundary values of x, and the control functions ul, u2, v have their values (u,u ) e U = E, v c V = (-1} U (13. Actually, we want to minimize I in the class Q of all systems (x,u,u2,v) with ul,u2 measurable in G, v measurable on r, and x any element of the Sobolev space W (G). We shall consider here the sets o 12 22 2 2 1 2 Q(lYx) [ [(z,z,z )lz > +q Ix +u1 +2 z = ul z = U2 (u u2) c E2] E, R=[(z0z) j>O, z v, v + 1 c E2 We have here r = 2, the sets Q have property (Q), or Q(3), in A = dr x E We have also r' = 1, the sets R have property Q(1) in B = r, have property (U), but they are not convex and do not have property (Q). In the search of the minimum of I in Q we can limit ourselves to those 31

elements (x,ul,2, v) E Q with I < M for some constant M. Here f =2 + 2 2 2 + x + U1 + u2, go = O, hence r = 0, J = O. We take z(t) = (x,x), y(t) = x, z(t) = Yx, Y(t) = O. If [Xk] is a minimizing sequence, hence lxkll, < N for 2 some constant N., there is a subsequence, say still [k] for the sake of simpli1 2 city, such that xk -t x weakly in W (G), zk I z weakly in (L2(G)), 2 k strongly in L2(G), k z strongly in L+ z(, y strongly in L2(r). Lower 2 k 2 k 2 closure theorem (4.1) may be applied witj p = 2, p' = 0. Example 2. Let us consider the problem of the minimum of the cost functional 2 2 2 2 2 2 I[X,U1,U2V] = ff(x +X +X +u +u2 (1-u2 ) d) d2 + f (Xx-l) ds 1 2 G r 1 2 2 F with differential equations x +x U1u + = U2 a.e. in G, yx = cos v, yx =sin v, s- a.e. on r = aG, where G and r are as in example 1, where yx denotes the boundary values of x, and the control functions u1,u2,v have their values (ul,u2) c U = E v E V = E. We want to minimize I in a class Q of systems (x,ul,U2, v) with Ulu2 measurable in G, v measurable on r, x any element of the Sobolev space 2 2 2 2 W2(G) satisfying an inequality x II 2+llx +x I2< M (Ma constant large 2 12 112 112 - enough so that 2 is not empty). We shall consider here the sets 32

o 1 2 o 2 2+ 2 2 2 2 1 Q(y) (z,z,z )1z, > y_ +y2 +y +u +2 (1-u2) Z z = 2ty [(20,21,22)( 20 51 2 3 1 2 2 z = u2, (ul,u) E ] c E, 2 1 2 o O 2 1 2 R(y)= [(z0,,z z2) > (Y_1), z = cos v, z = sin v, v E] cE3, where y = (y,Y2 Y3) in Q(y), and y in R(y) are arbitrary. Here we have r = 2, r' = 2. The sets Q have property Q(2), but they are not convex, and do not have property (Q), or Q(3). The sets R have property Q(1), but they are not convex, and do not have property (Q), or Q(3). They all have property 2 2 2 2 2 2 2 Q(O), or (U). Here f = Y1 + Y2 + Y3 + ul + u2 (1 - u ), go = (y-l), and 0 2 we can take t = 0, r = 0. We have here z(t) = (x~ + x, x~ + x ), y(t) (x, x, x ), z(t) = (yx5, yx), O(t) = yx. If [Xk] is any sequence xk E (x]Q, then there is a subsequence, say still [k], such that xk + x weakly in W2(G), (Zk) + (z) weakly in L2(G), z (Zk) + ( strongly in L2(G), Yk + y strongly in (L2(G))3, zk + strongly in L (r), and Yk strongly in L (r). Lower closure theorem (4.1) applies with p = 1 and p' = 0. 33

REFFERENCES 1. L. Cesari, Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I and II. Transactions of the American Mathematical Society. Vol. 124, 1966, pp. 369-412, 413-429. 2. L. Cesari, Existence theorems for optimal controls of the Mayer type. SIAM Journal on Control, Vol. 6, 1968, pp. 517-552. 3. L. Cesari, Existence theorems for multidimensional Lagrange problems. Journal of Optimization Theory and Applications, Vol. 1, 1967, pp. 87112. 4. L. Cesari, Existence theorems for abstract multidimensional control problems. Journal of Optimization Theory and Applications, Vol. 6, 1970, pp. 210-236. 5. D. E. Cowles, Upper semicontinuity properties of variable sets in optimal control. Journal of Optimization Theory and Applications. To appear. 6. L. Cesari and D. E. Cowles, Existence theorems in multidimensional problems of optimization with distributed and boundary controls. Archive for Rational Mechanics and Analysis. To appear. 7. E. J. McShane and R. B. Warfield, On Filippov's implicit functions lemma. Proceedings of the American Mathematical Society, Vol. 18, 1967, PP. 41-47. 34

8. C. B. Morry, Functions of several variables and absolute continuity. Duke Mathematical Journal, Vol. 6, 1940, pp. 187-215. 9. C. B. Morrey, Multiple integral problems in the calculus of variations and related topics. Univ. of California (N. S. ) Vol. 1, 1943, pp. 1-130.

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